Centre of mass of a lamina by integration

Understanding Centre of Mass by Integration

Integration in mathematics refers to a fundamental concept in calculus which is used to compute areas, volumes, central points, and many useful quantities. It is best understood in contrast and relation to differentiation, the inverse operation.
The process of finding the centre of mass by integration involves the division of the lamina into infinitesimally small particles, each with a small mass and position.
The position of the centre of mass of each particle is then multiplied by its mass and integrated over the entire object. This gives the total mass moment about the axis.
Sequence of the integration process is not important, integrating with respect to either x or y first will give the same result.

The general formulas to find the centre of mass (X,Y) of a lamina using integration are X = 1/M ∫ x dm and Y = 1/M ∫ y dm, where dm is the mass of an infinitesimal element of area dA at position (x,y), and M is the total mass of the lamina.
For a uniform lamina, the mass dm of an element dA is directly proportional to its area dA, with the mass per unit area (or surface density) as the constant of proportionality. Hence, dm = σ dA where σ is the surface density.
Hence the formulas can be rewritten as X = 1/A ∫ x dA and Y = 1/A ∫ y dA, where A is the total area of the lamina.
The limits of the integration depend on the geometry of the shape.

For example, to find the centre of mass of a uniform rectangular lamina of height h and base b, the limits of x would be 0 and b, and for y would be 0 and h. The centre of mass would come out to be X = b/2 and Y = h/2.
For a uniform semicircular lamina of radius r, the shape is more complex. We would have to express y or x in terms of the radius r and setup the integration accordingly.

This method of finding the centre of mass is particularly useful when the lamina is not of uniform density or when it has no lines of symmetry.
The centre of mass plays a crucial role in fields such as statics and dynamics. Understanding how to compute it and its implications can make solving related problems significantly easier.